To solve the problem of finding the Minimal Spanning Tree (MST) using the distances provided in the diagram, we can use an MST algorithm such as Prim's Algorithm or Kruskal's Algorithm. I'll describe the steps and then compute the solution.

Steps for Kruskal's Algorithm

1. List all edges with their weights.
2. Sort the edges in ascending order of their weights.
3. Start building the MST:
   • Add the smallest edge to the MST if it doesn't form a cycle.
   • Repeat until all vertices are connected and the MST contains $V-1$ edges (where $V$ is the number of vertices).
4. Calculate the total weight of the MST.

Steps for Prim's Algorithm

1. Start from an arbitrary vertex (e.g., $A$).
2. Select the smallest edge connecting the vertex to another unvisited vertex.
3. Repeat the process, always choosing the smallest edge that connects any visited vertex to an unvisited vertex.
4. Stop when all vertices are included in the MST.

Solution using Kruskal's Algorithm

1. List all edges

Edges (vertex pairs) and their distances:

• $A \to B = 50$
• $A \to C = 100$
• $A \to E = 50$
• $B \to C = 300$
• $B \to D = 90$
• $C \to D = 100$
• $D \to E = 150$
• $D \to F = 100$
• $E \to F = 50$

2. Sort the edges by weight

Sorted list of edges:

1. $A \to B = 50$
2. $A \to E = 50$
3. $E \to F = 50$
4. $B \to D = 90$
5. $A \to C = 100$
6. $C \to D = 100$
7. $D \to F = 100$
8. $D \to E = 150$
9. $B \to C = 300$

3. Build the MST

Start adding edges without forming cycles:

1. $A \to B = 50$
2. $A \to E = 50$
3. $E \to F = 50$
4. $B \to D = 90$
5. $A \to C = 100$

At this point, all vertices ($A, B, C, D, E, F$) are connected. Total edges = $6 - 1 = 5$.

4. Total Weight of the MST

The minimal total length is: $50 + 50 + 50 + 90 + 100 = 340 \, \text{km}$

Sequence of MST Edges

• $A \to B$
• $A \to E$
• $E \to F$
• $B \to D$
• $A \to C$

This is the MST that connects all cities with the minimal total distance of 340 km.

Would you like a detailed explanation of how cycles are avoided or how the algorithm works step-by-step? Let me know!

Related Questions:

1. How does Prim’s algorithm differ from Kruskal's algorithm in approach?
2. What is the time complexity of Kruskal's algorithm, and how is it optimized?
3. How would the MST change if the edge weights were altered?
4. Can MST algorithms handle directed graphs, and if not, why?
5. How is cycle detection ensured when constructing an MST?

Tip:

When using Kruskal’s algorithm, always sort edges first and use a union-find data structure to efficiently check and avoid cycles.